Tuesday, September 25, 2012

My
students think this is great fun. They have no idea they are exploring
linear functions or algebraic relationships. All they know is that these
problems make them think and they seem to like that.I usually
introduce algebraic thinking problems to third grade students during
our unit on multiplication and division. As you know, this topic does go
on for quite some time and it can get a little, dare I say, dull.
Algebraic reasoning problems give young students a chance to apply their
knowledge of basic math facts to fairly complex problems. Problems
like this inspire young minds and satisfy their need for a greater
challenge. My students are incredibly proud when they are able to solve
one of these math problems successfully.To
make things even more interesting, I ask my students to create their
own scale problems. We begin with two scales which I improvise with
pieces of plain copy paper. I then give the students a variety of
objects such as base ten blocks, colored cubes, and geometric tiles.
They choose two types of objects to work with and begin creating their
scale problems. They have to decide upon a value for each scale and then
check it to make sure it works. After that, the students switch places and try
to solve the problem. It's one of their favorite activities and it
gives me great joy to see them so actively engaged in problem solving.Give it a try. You won't be disappointed!

Tuesday, September 18, 2012

I designed the math game, Shape Mods, and the accompanying Transformation Workshop
to provide students with opportunities to practice geometric
transformations. The object of the game is to transform the starting green figure
into the final red figure using anywhere from one to four transformation
blocks. The blocks include translation, rotation about the origin, and
reflection across horizontal and vertical lines as well as y = x and y =
-x. Once the blocks are in place, students can watch the
transformations play out in the order they chose.

We've
been projecting Shape Mods on the whiteboard and having our 5th and 6th
grade classes engage in spirited competitions. Each team is given a set
of pink, white, blue, and yellow discs to represent vertices. This helps
students visualize the steps before locking in an answer. Students have
also enjoyed practicing independently.

Saturday, September 15, 2012

I generally work with students who would be considered above average in school. But every so often a student comes into my life
for whom each new math concept is an exhausting struggle. Math is an
endless menu of incomprehensible and unrelated steps to be memorized and
catalogued. That there could ever be any purpose to, let alone any joy in,
this cryptic jumble of numbers, formulas, and procedures is
unimaginable.

These are the students that
inspire creativity, awaken passion, and elicit reflection. They are the
reason I teach and they are the students who make
me want to be a better teacher.

Em came
to be my student last year at the start of grade 7. A portfolio of
sixth grade work revealed a math program that was largely focused on
computation. Em had countless examples of multiplication and division of
whole numbers, fractions, and decimals. While it was clear that Em
attempted to dutifully follow the algorithm of the day, there were signs
that something was seriously amiss.

Em
needed to prepare for a private school entrance exam. The test primarily
consisted of problem solving, pattern recognition, and general
mathematical concepts. Em could only confidently answer 2 of the 50
questions on the diagnostic test. As we worked our way through the
problem set, numerous content holes, flawed reasoning, and
misconceptions were exposed. Em had managed to mimic the computational
steps necessary to pass classroom tests and quizzes but had escaped any real
mathematical learning. Em did not understand place value, could not order
simple unit fractions, saw no relationship among equivalent fractions,
did not understand the purpose of a decimal point, and lacked number
sense.

Em and I worked together regularly
for 6 months without any significant progress. I thought I had tried
everything - visuals, manipulatives, real world examples, common
language, even acting out problems. Just as I was about to give up, the
connection I so desperately sought finally made an appearance.

We
were exploring decimals when Em called the decimal point a period. While
privately lamenting this student's misunderstanding, I wondered if
perhaps there might be something to it. I asked Em to explain further. Em
went on to tell me that the decimal point marks the end of the whole
numbers and the start of the "smaller pieces", the pieces that weren't
quite whole yet. In Em's mind, that was very similar to the way a period
ends one thought but can also signal the start of a new one. From there, Em
told an elaborate tale of a fantastical world of whole numbers and
pieces, how they are kept apart by the will of the decimal point, how
the wholes and pieces organize themselves into groups by size, and how
these groups are either 10 times bigger or 10 times smaller than groups
on either side. Em also described how the decimal can make numbers grow
or shrink by moving its location and that it is always present even when
there are no "pieces".

Em
understood decimals better than any 7th grade student I had ever met.
No standardize test in the world would ever ask Em to tell the story of
the wholes and the pieces. Yet, that was the only way Em could
confidently share her knowledge.

Since that time, classes with Em have been rather
magical journeys into far-off lands where numbers and symbols come to
life and tell their stories. Em is in 8th grade today and is struggling
with the rigidity of her pre-algebra course. Her class has been studying
the distributive property and combining like terms. Em confided
that she just didn't get it. I mentioned something about helping the
expression escape from its parentheses prison. Before long, Em had
crafted a story about the number guard that stood watch outside the
prison, the banning of subtraction, and the look-alike law.

Is it easier to solve? It's exactly the same
problem, isn't it? The visual representation of the problem makes a huge
difference, though. Now it's obvious that the two green blobs are the same size
even though they exist on different scales (or different equations).
And the two blue blobs are the same. Aha, the two red ones as well.

I
have prealgebra and algebra students who ask if the A in one
equation has the same value as the A in another, whether the two
equations are part of a system or not. Imagine if those students had
been solving math puzzles like this one throughout elementary school.
Would their concept of variable be more clear?

Elementary
age students at the math center think problems like this are great fun.
They have no idea they are exploring linear functions or algebraic
relationships. All they know is that these problems make them think.
Algebraic reasoning problems give young students a chance to apply their
knowledge of basic math facts within fairly complex scenarios. Problems
like this one inspire young minds and satisfy their need for a greater
challenge. Our students are incredibly proud when they are able to solve
one of these math problems successfully.

How
would a young student solve such a problem? We ask our students to
compare any two scales and find what they have in common. Let's take the
first two scales. Students will point out that the blue blob is common
to both. Then we have them look for differences. Students notice that
the scale weights differ and the partner blobs are different. We ask
them what they think might be causing the weight on the second scale to
be greater. It's obvious to students that the bigger red blob on scale
two is causing an increase in weight.

Once
they understand the effect of changing the partner from a small green
blob on the first scale to a bigger red blob on the second, we can look
at the quantitative aspects of the problem. We then ask what is the
difference in weight between the two scales. Students will do the
computation and find the difference is 19. That's the difference between
scale one and scale two. What else does this number mean? What other
difference does it describe? Students will relate 19 to the difference
in weight of the green blob and the red blob. The red blob weighs 19
more weight units than the green blob.

We write this as:

R = G + 19

Now
we have a relationship between the red blob and the green blob. This
relationship tells us that if we replace a red blob with a green blob
plus 19 weight units the scale reading will stay the same. So let's do
it.

We head over to third scale. The equation there is:

G + R = 33

We replace the red blob. We get:

G + G + 19 = 33

What if we take 19 weight units off the scale?

What will the scale read then?

33-19 = 14

Our new equation is:

G + G = 14

At
this point, students recognize this as a doubles problem and easily
find the value of G to be 7. They then use this value to find the
weights of the other blobs.

Modeling this
problem with young students as a whole group activity is a very
powerful. They excitedly share their insights and answers. We'll do
several of these together before they work independently to solve
similar problems. Eventually we make our way toward the original
abstract problem. We replace the green blob with the letter G, then the
blue blob disappears and gives way to the letter B, and finally we part
ways with the red blob and bring out the letter R. The letters remain on
the scale however so the context is reserved. Once the students are
comfortable working with letters, we then remove the scale. Students
solve systems of three equations by the end of 5th grade. More importantly,
students learned how to think through abstract problems, a skill that
will forever be of value.

Monday, September 10, 2012

While foraging for markers, a student in one of my math and programming classes
stumbled upon some old science equipment I keep in the closet. The
air-propelled rocket launcher was promptly brought out of retirement and
set up in the long rectangular space at the rear of the math center. It
wasn't long before a rousing game of "hit the target" was underway.
Based on the number of times the soft foam rocket came perilously close
to my head, it would seem I was the target although everyone agreed they
were aiming for the algebra poster.

Sensing an opportunity among
the chaos, I grabbed a hula hoop out of the closet of science and
placed in on the floor on the other side of the room. The
hula hoop proved a more interesting target and it wasn't long before the
discussion headed in the direction of angles and velocity. An impromptu
lesson on projectile motion ensued.

We measured launch angles
and landing distances and recorded flight times. We refined our
understanding of velocity and used horizontal motion data to find
starting velocities. Through our experiments, we hit upon combinations
of velocities and launch angles that would land our rocket inside the
hula hoop. One student, who grew frustrated with the trial and error
process, asked,

"Can we calculate the velocity and angle if we know where we want the rocket to land?"

Since
this was a math and programming course, I suggested we write a program
that models the experiment and perhaps make a game based on this
student's question. After several weeks of brainstorming, coding, revising, and experimenting, we came up with this:

There are both elements of game design (points
awarded, incentives for calculations and good guesses, increasingly
difficult levels, and interesting sound effects) and elements of
instructional design (timed flights, recorded data, and explanation of
math equations). We also replaced the theme of destruction typically
seen in these games with a more positive rescue mission plot. I
am incredibly proud of this group and all that they have accomplished
this year. I know some will be moving on but I hope we can continue our
work next year.

Friday, September 7, 2012

A seventh grade student came to the math center to prepare for
a test on fractions. She brought in a review sheet with various
practice problems which she completed with time to spare. The student,
somewhat anxious about the test, asked if I could make up problems on
the whiteboard. I complied and wrote out the following problem:

My student proceeded to simplify by canceling common factors.

And then declared this was all she could do.Me: I think you can simplify this further.The student tested adjacent numerator-denominator pairs: 24 and 7, 5 and 8, 8 and 27Student: No, that's it. I can't simplify this.Me: Have you tried 24 and 27?Student: (spins around) I wanted to but aren't they too far apart?Me: Nah, I think the limit is around 3 feet.(I couldn't resist.)After 45 minutes of rather predictable practice problems, we finally had a teachable moment.

It
makes me wonder what other misconceptions our students have that we
never uncover. And it reminds me to never stop pushing past the surface.

Tuesday, September 4, 2012

A student in my middle level programming course brought in a word problem from school.

"The
Billy Bonkers candy factory is having a contest. The candy makers
placed a silver ticket in every 600th chocolate bar and a golden ticket
in every 720th chocolate bar. Anyone who purchases a chocolate bar
containing both tickets wins the grand prize. If 10,000 chocolate bars
are sold, how many grand prize winners will there be?"

Once
we determined this was a Least Common Multiple problem, we talked about
various ways to solve it. One student suggested writing out the first
few multiples of each number and looking for common numbers. Another
student had learned about factor trees and knew how to use prime factors
to find the LCM. Yet another student showed us how to use a Venn
diagram to organize the prime factors.

While
discussing the relative efficiency of each method, one student
declared, "It would take forever to do this if we had a lot of numbers."

"It
sure would seem that way," I thought to myself, relishing the near
perfect segue this statement introduced. And before I could get the
words out, another student asked if we could write a program.

"We certainly could try," I said aloud.

We
began by attempting a simpler problem: finding a common multiple of two
numbers; not necessarily the smallest one. Everyone agreed that the
easiest method was to multiply the numbers together. We then compiled a
list of products and LCMs for various pairs of numbers. I wanted the
students to find a connection between the product and the LCM. How do
the divisors relate to the original number pairs?

The
connection eluded the students and it took quite a few hints to help
them see that the product of two numbers is related to their LCM by the
greatest common factor. We had the beginnings of an efficient algorithm.

LCM = (number1 x number2)/GCF

Or
did we? Had we just shifted the difficulty in finding the LCM to the
similarly difficult task of finding the GCF? Is there a quick way to
find the GCF of two numbers?

We talked
through some known methods (listing the factors of each number, using
prime factorization, applying the venn diagram) and opted to look for a
better way.

We uncovered some interesting
facts about the GCF. It's never larger than the smaller number or the
difference between the two numbers. Consecutive numbers always have a
GCF of 1 making them relatively prime. One student suggests an algorithm
that tests if each number, from the smaller of the pair down to 1,
evenly divides both numbers. The first number that worked would be the
GCF. We look at the numbers 6 and 20. The program would test 6, 5, 4, 3,
2, 1. The number 2 evenly divides 6 and 20 and would be the GCF. I
asked what would happen if the two numbers were as large as the numbers
in the word problem - 600 and 720. Is it really necessary to test all
the numbers from 600 to the GCF? Someone suggests testing only the
factors of 600. Nice! But how do we find all the factors of 600? And
even if we had an algorithm for this, we'd have to test several factors
before hitting upon the GCF.

Time to
introduce Euclid's algorithm. We divide the larger number by the smaller
number and find the remainder. If the remainder is zero, the smaller
number is the GCF. If not, we continue the process. During each
iteration, the original smaller number becomes the dividend and the
remainder becomes the divisor. When we finally get a remainder of zero,
the last divisor is the GCF.

Finding the GCF of 600 and 720 became an astonishingly easy task.

720/600 = 1 r 120

600/120 = 5 r 0GCF = 120

The
students were amazed! Two steps. That was all. After trying a few more
number pairs, the class was satisfied that the procedure was foolproof.
Back to the original problem; finding the LCM of a group of numbers. The
equation we had discovered earlier was:

LCM = (number1 x number2)/GCF

The
plan was to take the first two numbers in the set, apply Euclid's
algorithm to find the GCF, then use the GCF to divide the product of the
two numbers. We would repeat the procedure with the current LCM and the
next number in the set until we reached the final number and,
therefore, the final LCM.Naturally, this begs the question, "What can we do with this?", which I will ask the class at our next meeting.

About the Author

Colleen King is a math educator with 15 years experience working with K-12 students in a variety of settings. Colleen publishes MathPlayground.com and develops math games, teaching tools, and learning resources that are used in classrooms throughout the world. She has presented her work at ISTE and NCTM conferences and has co-authored several teaching articles.